# In a CRR model, find the Initial investment of the hedging strategy

Given a Cox-Ross-Rubinstein model with $$T=10$$, $$u=1.1$$, $$d=0.9$$, $$r=0.02$$, $$S_0=100$$ and a European call option with Strike $$K=220$$, find the initial investment of the hedging strategy.

I know how to compute the fair price at $$t=0$$ for the European call, say it is $$x$$, and I know that for the initial investment $$(\alpha_0,\beta_0)$$ of the Hedging strategy where $$\alpha_0$$ is the weight on the stock and $$\beta_0$$ on the bond, we have $$x=100\alpha_0+\beta_0$$. But there needs to be another equation such that I can solve it.

Another possibility, which it seems like is the one you're looking for, is finding $$(\alpha_0, \beta_0)$$ themselves. In finding the price of the option at time 0, you are effectively finding these also. In particular, you can explicitly solve: $$100u\alpha_0 + (1 + r) \beta_0 = \text{Value of the option at t=1 if the stock goes up}$$ $$100d\alpha_0 + (1 + r) \beta_0 = \text{Value of the option at t=1 if the stock goes down}$$
And note that there should be no inconsistency with the equation you have provided, $$100 \alpha_0 + \beta_0 = x$$.